Open-domain Relational Triplet Extraction (ORTE) is the foundation for mining structured knowledge without predefined schemas. Despite the impressive in-context learning capabilities of Large Language Models (LLMs), existing methods are hindered by their reliance on static, heuristic-driven prompting strategies. Due to the lack of reflection mechanisms required to internalize erroneous signals, these methods exhibit vulnerability in semantic ambiguity, often making erroneous extraction patterns permanent. To address this bottleneck, we propose a Knowledge Reconstruction-driven Prompt Optimization (KRPO) framework to assist LLMs in continuously improving their extraction capabilities for complex ORTE task flows. Specifically, we design a self-evaluation mechanism based on knowledge restoration, which provides intrinsic feedback signals by projecting structured triplets into semantic consistency scores. Subsequently, we propose a prompt optimizer based on a textual gradient that can internalize historical experiences to iteratively optimize prompts, which can better guide LLMs to handle subsequent extraction tasks. Furthermore, to alleviate relation redundancy, we design a relation canonicalization memory that collects representative relations and provides semantically distinct schemas for the triplets. Extensive experiments across three datasets show that KRPO significantly outperforms strong baselines in the extraction F1 score.